Then restore My database from a backup - but make sure to run DBCC CHECKDB on that very backup to ensure it's not corrupt itself.

DBCC CHECKDB with REPAIR_ALLOW_DATA_LOSS is the very last resort . Before that, you should restore a valid backup. Once you do use REPAIR_ALLOW_DATA_LOSS, don't expect your database to magically come back to life again - most likely, irreparable damage has occurred - especially in this case since it complains about system table corruption.